Isn't AirPlay only good for screen casting as opposed to casting videos directly? My conclusion was that it's not feasible to use AirPlay to cast a video at the full frame rate and sound synced.
AirPlay can do both. There's a "screen mirroring" button in Control Center that streams your phone screen to the TV, but if you tap the AirPlay button on a video player or in the audio device selector, the TV will stream the video directly from the server at native resolution and FPS, without going through your phone (you can turn your phone off and playback will continue).
Depends on if the video source itself supports Airplay or not. Most don't.
AirPlay or more accurately "screen mirroring" does cast sound and more often than not recognizes the video content and casts it full screen (from most iOS media apps such as youtube at least). It doesn't always work on my samsung TV without an Apple TV device though, in about 10% of cases it'll just fail to connect to the TV altogether
You can cast video e.g. from the QuickTime app or a <video> tag in the browser too which won’t just mirror your screen. In fact the cast video won’t even show on your device’s screen but only on the receiver in that case.
I use an app called Airflow on my Mac to stream local video files to Apple TV. It's $20, but lifetime license and it's been working fine through the years.